{"id":"8430fda1-1bef-482c-ad80-5e1ba576f2fb","arxiv_id":"2412.04110","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":11,"one_line_summary":"Fine-tuning Llama-3.1-8B to generate Prolog programs from a fixed 54-operator library reportedly reaches 84.8% on MATH counting/probability, but test-set solutions were added to training, so the headline is not a clean held-out result.","lead":"We propose and test a method that makes large language models solve math story problems by writing Prolog code using a fixed library of 54 background math operators. On the MATH counting and probability problems, a fine-tuned Llama-3.1-8B model reportedly reaches 84.8% test accuracy, but the evaluation protocol mixes test-set solutions into training, so the headline number is inflated.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 84.8% test-set result is in-sample: Section 3.2 adds derived test solutions to the training set and reruns self-training, so the test set is no longer unseen when accuracy is reported; the central generalization claim is unsupported.","rationale":"The central empirical claim is that cross-validated self-training discovers correct Prolog solutions for previously unseen problems, quantified by 84.6% and 84.8% accuracy. The protocol in Algorithm 1 and the description in Section 3.2 show that the evaluation sets themselves are fed back into the training set via verified generated solutions. Under those conditions, the metric no longer measures generalization to unseen problems; it measures whether the model can emit a correct solution for a question it has already seen during fine-tuning. This is a correctness risk, not a stylistic disagreement, because the abstract and conclusion explicitly claim 'previously unseen problems' and 'unseen questions'. The 84.8% figure is explicitly obtained after augmenting test solutions into training, so it cannot support that claim. The 5-fold cross-validation suffers from the same issue if evaluation is performed on a fold after correct generated solutions from that fold have been added to D; the paper does not specify that evaluation happens before augmentation. Thus the strongest claim is unsupported by the reported numbers. I do not question the honesty of the authors or the potential utility of the MATH-Prolog corpus; the dataset and background-operator idea may be valuable, and a clean held-out evaluation could rescue the claim. But as written, the reported accuracy is an in-sample measure. This matches the reader's weakest-assumption analysis, so the verdict should remain REJECT. A resubmission with a truly held-out test split, no test-set augmentation, and per-fold evaluation before augmentation would make the claim CONDITIONAL at worst.","tokens_in":11128,"tokens_out":7361,"duration_ms":71884,"concrete_test":"Use only the 625 training samples as D. Run self-training without ever adding test-set questions or solutions to D; at the end, sample one solution per test question from the final model and compute accuracy. Also, for each CV fold, save the model checkpoint immediately before any solution from that fold enters D; evaluate that checkpoint on the fold. Compare these held-out accuracies to the reported 84.8%/84.6%. If the held-out figures are materially lower (e.g., near the 71.1% no-predicate baseline), the claimed generalization is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim requires a held-out test set. Algorithm 1 adds every verified generated solution to D, including solutions for the fold being used as the generation set. Section 3.1's metric sums over all folds, but it does not state whether the prediction is sampled before or after the question's solution enters D. If sampled from the final model, the fold questions have been in D since their first correct solution was found, so the 84.6% is not held-out. For the test set, the paper is explicit: after first-round self-training, 'we used the derived test solutions ... and augmented them into the training set and rerun self-training again', then reports 84.8% test accuracy. That number is necessarily in-sample. Even the 75.7% first-round test accuracy is suspect because the same loop adds correct test solutions to D during the run; unless evaluation is performed exclusively before any test-set sample is added, the test set is not unseen. Therefore the reported numbers do not establish the abstract's claim of solving 'previously unseen problems'.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes representing mathematical reasoning problems as Prolog programs built from a fixed set of 54 hand-crafted background predicates, and introduces MATH-Prolog, a corpus derived from the counting and probability categories of the MATH dataset. The authors fine-tune Meta-Llama-3.1-8B-Instruct with LoRA and use a 5-fold cross-validated self-training procedure (Algorithm 1) that iteratively generates Prolog solutions for a held-out fold, verifies them with SWI-Prolog, and adds correct ones to the training set. They report 84.6% accuracy on the cross-validated set and 84.8% on the test set, and claim that the approach solves previously unseen problems and that including background operators in prompts improves solution coverage.","tokens_in":11412,"tokens_out":7188,"duration_ms":71462,"significance":"If the evaluation were sound, the paper would make a useful contribution: a structured, mechanically verifiable code-generation approach to competition-level counting and probability problems, together with a curated corpus and an interpretable graph-of-predicates representation. The corpus curation is substantial work, and the diversity examples in Appendix A.1 are illuminating. However, the central experimental claims are not trustworthy as reported. Both headline accuracies are computed after the evaluated data have been added to the training set, so the numbers cannot support the abstract's assertion that the model solves 'previously unseen problems.' With a clean held-out evaluation, the approach might still be valuable as a data-augmentation technique, but that claim is not established by the current manuscript.","major_comments":[{"comment":"The pseudocode explicitly adds every verified generated solution from the generation set G into the training set D (line 'Add (q,ˆa) into S and D'), and subsequent epochs fine-tune on D. In the 5-fold protocol, each fold is used as G with the remaining folds as D, so by the time the model is scored on a fold, correct generated solutions from that fold have already been incorporated into training. The 84.6% accuracy reported in Section 3.2 is therefore not an out-of-sample measure, and the claim of solving 'previously unseen problems' is unsupported by this experiment.","section":"Section 2.2, Algorithm 1"},{"comment":"The test-set result is explicitly in-sample. The text states: 'When we used the derived test solutions from the first-round self-training and augmented them into the training set and rerun self-training again, the test set accuracy was increased to 84.8%.' Augmenting the test solutions into the training set before reporting test accuracy means the test set is no longer held out. The same concern applies to the first-round 75.7% test accuracy: unless evaluation is performed before any test-set sample is added to D, which the manuscript does not state, the model has seen test-set solutions during training. The 84.8% number must be retracted or reframed as an in-sample data-augmentation accuracy.","section":"Section 3.2"},{"comment":"The accuracy metric (unnumbered equation under 'Evaluation') sums over folds and divides by |D|, but it does not specify which model checkpoint is used to produce spred for a fold, nor whether predictions are drawn before or after that fold's correct generated solutions are added to D. Without this specification, the metric cannot be interpreted as measuring the ability to discover solutions to unseen problems. The authors should either (i) evaluate each fold with a model trained only on the other folds and without any fold-generated samples in the training set, or (ii) clearly separate a discovery set S (solutions found) from the training set D and report accuracy computed only on checkpoints before any sample from the evaluated fold is added to D.","section":"Section 3.1, Evaluation"}],"minor_comments":[{"comment":"The accuracy equation uses |D| in the denominator but the sum is over folds D_i; if D denotes the union of all folds, the notation should be defined explicitly.","section":"Section 3.1"},{"comment":"The text claims 54 background operators, but Appendix A.2 appears to list fewer; the appendix should enumerate all 54 operators or clarify the count.","section":"Section 2 vs. Appendix A.2"},{"comment":"Figure 3 is referenced but not shown in the provided text, and its axes and experimental protocol are not described; the reader cannot verify the learning trajectories or the 84.6% value from the figure.","section":"Section 3.2 and Figure 3"},{"comment":"The reference 'Abhimanyu Dubey et. al. 2024' is malformed; the citation should use the standard author–year format with the full author list or 'et al.' as appropriate.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper may be salvageable if the experimental evaluation is redone with a strict held-out protocol. The current version's headline accuracies are computed in-sample, so the main claims are not supported. I would ask the authors to (a) report pre-augmentation accuracy for each fold, (b) clarify exactly when predictions are sampled relative to when generated solutions enter D, and (c) either remove the 84.8% test-set accuracy or reposition it explicitly as a data-augmentation result rather than a generalization result. If the authors cannot provide a clean held-out evaluation, the generalization claims should be withdrawn."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead this one so you know what's out there: it curates a new dataset (MATH-Prolog) of Prolog solutions for counting/probability problems from MATH, using a fixed set of 54 background operators, and fine-tunes an 8B Llama to generate such solutions. That's genuinely new and likely useful for the neural-symbolic community. The operator decomposition idea (combination as permutation + factorial + division, etc.) is a nice way to encourage diverse solutions.\n\nBut the evaluation doesn't support the headline. The 84.8% test accuracy is obtained after augmenting verified test solutions into the training set and rerunning self-training. That's in-sample, not generalization. Same for the 84.6% cross-validated number: Algorithm 1 adds correct generated solutions for the held-out fold into D, and the model is then trained on them; the final metric measures recall of the discovered solutions, not ability on unseen problems. Even the 75.7% first-round test number is tainted because correct test solutions were added to D during the run.\n\nWhat's missing is a clean held-out evaluation: train on the folds only, sample for the held-out fold before any of its solutions enter D, and report that accuracy. The paper doesn't give that. There are also no error bars or multiple seeds, and the prompt ablation is a single comparison.\n\nThe problem is in the measurement, not the math. The corpus work is real, the paper is transparent about its procedure, and the citations look fine. The flaw is fixable: report the one-shot accuracy on an untouched test split, or at least separate the discovery rate from the memorization rate.\n\nFor you: if you work on math reasoning or self-training with verifiable code, the dataset is worth a look. The paper as written should not be accepted for its stated claims. I'd send it to review only to force a clean evaluation; a careful referee would reject or demand major revision. My recommendation: treat the 84.6/84.8% numbers as in-sample and discount them. The idea, not the numbers, is the contribution.","headline":"New dataset and operator framework, but the reported accuracies are in-sample because the evaluation adds generated solutions from the target set into training.","tokens_in":11900,"tokens_out":4712,"would_cite":false,"duration_ms":45367,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Fine-tuning an 8B LLM to emit Prolog built from 54 standardized math predicates reaches 84.8% accuracy on competition counting and probability problems.","keywords":["mathematical reasoning","Prolog","background operators","self-training","large language models","MATH corpus","verifiable reasoning","counting and probability"],"falsifier":"Run the same 5-fold procedure but withhold every generated solution from the training set until after the fold has been scored, and compare the accuracy; if it drops well below 84.6%, the reported gain comes from training on the test problems themselves.","tokens_in":10899,"feed_emoji":"🧮","tokens_out":10554,"duration_ms":85442,"temperature":0.7,"pith_summary":"This paper argues that competition-level math reasoning can be recast as constraint search in Prolog, and that a large language model can be taught to write such programs if it is given a fixed vocabulary of background mathematical predicates. To test this, the authors curate MATH-Prolog, a corpus of 625 hand-corrected Prolog solutions for counting and probability problems from the MATH benchmark, and fine-tune Meta-Llama-3.1-8B-Instruct on it. They then apply a 5-fold cross-validated self-training loop in which the model samples new solutions for the held-out fold, keeps only those whose output matches the reference answer when executed by SWI-Prolog, and adds them to the training data. The claim is that this procedure discovers diverse, correct, fully computable solutions for previously unseen problems, reaching 84.6% cross-validated coverage and 84.8% accuracy on the test set after augmenting the training set with derived test solutions. The reason to care is that, unlike chain-of-thought, the generated reasoning is machine-executable and verifiable, which could make LLM math answers auditable.","feed_headline":"LLM math accuracy hits 84.8% with Prolog building blocks","feed_subtitle":"Standardized math predicates make an 8B model's reasoning steps executable and checkable on competition problems.","key_machinery":"The load-bearing object is the set of 54 hand-crafted background mathematical predicates (e.g., combination(N,R,C), factorial(N,F), findall/3) that standardize every Prolog solution as a composed computation graph. These operators both constrain the model's output space and expose decompositional equivalences (for instance, combination can be rewritten as permutation composed with factorial and division), which the self-training loop exploits to generate diverse but still correct programs. The second piece of machinery is the K-fold cross-validated self-training algorithm: for each fold, the model samples up to J solutions per held-out question, executes each with SWI-Prolog, and adds any solution whose numerical result matches the reference into the training set before the next epoch. This is what turns a static corpus into an expanding one, and it is also the step that makes the reported accuracy scores sensitive to whether generated test-set solutions are reused as training data.","core_discovery":"The central claim is that declarative Prolog programming with a standardized set of background operators is a better target representation for LLM math reasoning than free-form natural language or procedural Python. The authors define 54 mathematical predicates such as combination/3, factorial/2, findall/3, probability_of_event/4, and cardinality/2, and require every Prolog solution to compose these operators, so a solution is effectively a directed computation graph from problem-specific facts to the answer. Fine-tuning an 8B Llama model on the resulting MATH-Prolog corpus makes it possible to generate such graphs for unseen questions, and the cross-validated self-training algorithm samples multiple candidate programs, executes them with SWI-Prolog, and keeps only those that evaluate to the correct final answer. The paper reports that this yields 84.6% coverage on the 5-fold cross-validation and 84.8% accuracy on the test set, and that including the background operators in the prompt improves the learning trajectory and solution coverage.","pith_inferences":["Editorial inference: because Algorithm 1 adds verified generated solutions to the training set before scoring, the reported test-set accuracy of 84.8% may partly reflect the model's exposure to the test problems' target programs; an independent held-out evaluation would be needed to separate memorization from generalization.","Editorial inference: the background-operator vocabulary functions as a soft grammar constraint that shrinks the space of outputs the model must explore, so part of the gain may come from the restricted output space rather than from Prolog semantics per se.","Editorial inference: the decompositional operator structure suggests a cheap route to more training data, namely composing operators into new problems or rewriting existing solutions at different levels of decomposition, without further manual labeling."],"forward_implications":["If a fine-tuned 8B model can generate executable Prolog at 84.8% coverage, then competition-level counting and probability problems can be delegated to a verifiable search process rather than to uncheckable verbal reasoning.","The cross-validated self-training loop removes much of the manual curation burden: correct solutions are discovered and verified automatically by execution, which is why the authors could skip hand-labeling the test set.","Because each generated solution is a computation graph over the 54 operators, the set of possible solutions is compositional: decomposing operators (e.g., combination into permutation, factorial, division) yields structurally different but equivalent programs that can serve as diverse training data.","Including the background operators in the input prompt accelerates the learning trajectory and improves final coverage, suggesting that exposing the vocabulary is itself an effective inductive bias."],"supporting_citations":[{"why":"Supplies the MATH benchmark, from which the counting and probability problems in MATH-Prolog are drawn.","marker":"(Hendrycks et al., 2021)"},{"why":"Provides the method of using GPT-4 to generate Prolog code for arithmetic reasoning, which the authors adapt to generate initial solutions for MATH-Prolog.","marker":"(Yang et al., 2024)"},{"why":"Inspires the self-training procedure used to automatically generate and select new Prolog solutions on the test set.","marker":"(Wang et al., 2024)"},{"why":"Defines the Alpaca instruction prompt format used to frame Prolog generation as an instruction-following task.","marker":"(Taori et al., 2023)"},{"why":"Supplies LoRA low-rank adaptation, the efficient fine-tuning method used to train the 8B model on a single GPU.","marker":"(Hu et al., 2022)"},{"why":"Defines the Meta-Llama-3.1-8B-Instruct base model that the paper fine-tunes.","marker":"(Dubey et al., 2024)"}],"fun_headline_variants":["Prolog graphs beat free text for 8B LLM math reasoning","84.8% on MATH: checkable Prolog solutions from Llama-3.1-8B","Executable math: self-trained Llama-3.1-8B hits 84.8%","Background operators turn LLM math into checkable Prolog","84.8% accuracy with verified Prolog solutions for MATH"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that scoring a fold after adding that fold's verified generated solutions to the training set still measures the model's ability to handle previously unseen problems, rather than its ability to reproduce memorized solutions.","fun_headline_variants_meta":{"raw":{"variants":["Prolog graphs beat free text for 8B LLM math reasoning","84.8% on MATH: checkable Prolog solutions from Llama-3.1-8B","Executable math: self-trained Llama-3.1-8B hits 84.8%","Background operators turn LLM math into checkable Prolog","84.8% accuracy with verified Prolog solutions for MATH"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000517,"raw_usage":{"total_tokens":2513,"prompt_tokens":954,"completion_tokens":1559,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":1453}},"tokens_in":570,"tokens_out":1559,"duration_ms":10387,"temperature":1.0,"reasoning_tokens":1453,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:45:31.072848+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the same 5-fold procedure but withhold every generated solution from the training set until after the fold has been scored, and compare the accuracy; if it drops well below 84.6%, the reported gain comes from training on the test problems themselves.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Inspires the self-training procedure used to automatically generate and select new Prolog solutions on the test set."}],"review_version":1}